MATH 251
Instr. K. Ciesielski
Spring 2003
SAMPLE TEST # 4
Solve the following exercises. Show your work.
Ex. 1. Let F(x, y , z) = ln(xy)i+(x+ sin z)j+(2y−2z)k. Calculate div Fand curl F.
Ex. 2. Evaluate C
xy ds, where Cis the parametric curve for which x=3t,y=t4, and
0≤t≤1.
Ex. 3. Evaluate the integral, where Cis the graph of y=x3from (−1,−1) to (1,1).
C
y2dx +xdy =
Ex. 4. Determine if the following vector field is conservative. Find potential function for a
field, if it is conservative.
(a) F=x3+y
xi+(y2+lnx)j
(b) F=(ycos x+lny)i+x
y+eyj
Ex. 5. Evaluate the integral
(π,π)
(π/2,π/2)(sin y+ycos x)dx + (sin x+xcos y)dy =
Ex. 6. Apply Green’s theorem to evaluate the following integral, where the simple closed
curve Cis the boundary of the circle x2+y2=1.
C(sin x−x2y)dx +xy2dy =
Ex. 7. Find the area of the surface of the graph of z=x2+ythat lies above the triangle in
the xy-plane with vertices at (0,0), (1,0), and (1,1).
Ex. 8. Evaluate
(a) 4
1−15
132
03−2
−3
03−2
31−1
−10 11
=
(b)
1−10
203
12−1
03
70
12
=
Ex. 9. Find the inverse matrix of
102
310
111
1